The K-partitioning problem consists in partitioning the nodes of a complete graph G = (V, E) with weights on the edges in exactly K clusters such that the sum of the weights of the edges inside the clusters is minimized. For this problem, we propose two node-cluster formulations adapted from the literature on similar problems as well as two edge-representative formulations. We introduced the first edge-representative formulation in a previous work [4] while the second is obtained by adding an additional set of edge variables. We compare the structure of the polytopes of the two edge-representative formulations and identify a new family of facet-defining inequalities. The quality of the linear relaxation and the resolution times of the four formulations are compared on various data sets. We provide bounds on the relaxation values of the node-cluster formulations which may account for their low performances. Finally, we propose a branch-and-cut strategy, based on the edge-representative formulations, which performs even better.

中文翻译：

---

K-分区问题：公式和分支和切割

K-partitioning 问题在于将完全图 G = (V, E) 的节点与恰好 K 个簇中的边上的权重进行划分，使得簇内边的权重之和最小。对于这个问题，我们提出了两个从类似问题的文献中改编的节点集群公式以及两个边缘代表公式。我们在之前的工作 [4] 中引入了第一个边缘代表公式，而第二个是通过添加一组额外的边缘变量获得的。我们比较了两个边缘代表公式的多胞体的结构，并确定了一系列新的刻面定义不等式。在各种数据集上比较了四种配方的线性弛豫质量和分辨率时间。我们提供了可能解释其低性能的节点集群公式的松弛值的界限。最后，我们提出了一种基于边缘代表公式的分支和切割策略，它的性能甚至更好。